Posts Tagged ‘mechanical power transmission’

The Angle of Wrap Formula

Wednesday, July 5th, 2017

    Last time we introduced a scenario involving a hydroponics plant powered by a gas engine and multiple pulleys.   Connecting the pulleys is a flat leather belt.  Today we’ll take a step further towards determining what width that belt needs to be to maximize power transmission efficiency.   We’ll begin by revisiting the two T’s of the Euler-Eytelwein Formula and introducing a formula to determine a key variable, angle of wrap.

 The Angle of Wrap Formula

The Angle of Wrap Formula

   

    We must start by calculating T1, the tight side tension of the belt, which is the maximum tension the belt is subjected to.   We can then calculate the width of the belt using the manufacturer’s specified safe working tension of 300 pounds per inch as a guide.   But first we’ll need to calculate some key variables in the Euler-Eytelwein Formula, which is presented here again,

T1 = T2× eθ)                                                       (1)

    We determined last time that the coefficient of friction, μ, between the two interfacing materials of the belt and pulley are, respectively, leather and cast iron, which results in a factor of 0.3.

    The other factor shown as a exponent of e is the angle of wrap, θ, and is calculated by the formula,

θ = (1802α) × (π ÷ 180)                                     (2)

    You’ll note that this formula contains some unique terms of its own, one of which is familiar, namely π, the other, α, which is less familiar.   The unnamed variable α is used as shorthand notation in equation (2), to make it shorter and more manageable.   It has no particular significance other than the fact that it is equal to,

α = sin-1((D1 – D2) ÷ 2x)                                       (3)

If we didn’t use this shorthand notation for α, equation (2) would be written as,

θ = (1802(sin-1((D1 – D2) ÷ 2x))) × (π ÷ 180)      (3a)

That’s a lot of parentheses!

    Next week we’ll get into some trigonometry when we discuss the diameters of the pulleys, which will allow us to solve for the angle of wrap.

 

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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What Belt Width does a Hydroponics Plant Need?

Friday, June 23rd, 2017

    Belts are important.  They make fashion statements, hold things up, keep things together.   Today we’re introducing a scenario in which the Euler-Eytelwein Formula will be used to, among other things, determine the ideal width of a belt to be used in a mechanical power transmission system consisting of two pulleys inside a hydroponics plant.   The ideal width belt would serve to maximize friction between the belt and pulleys, thus controlling slippage and maximizing belt strength to prevent belt breakage.

    An engineer is tasked with designing an irrigation system for a hydroponics plant.   Pulley 1 is connected to the shaft of a water pump, while Pulley 2 is connected to the shaft of a small gasoline engine.

What Belt Width does a Hydroponics Plant Need?

What Belt Width does a Hydroponics Plant Need?

   

    Mechanical power is transmitted by the belt from the engine to the pump at a constant rate of 4 horsepower.   The belt material is leather, and the two pulleys are made of cast iron.   The coefficient of friction, m, between these two materials is 0.3, according  to Marks Standard Handbook for Mechanical Engineers.   The belt manufacturer specifies a safe working tension of 300 pounds force per inch width of the belt.   This is the maximum tension the belt can safely withstand before breaking.

    We’ll use this information to solve for the ideal belt width to be used in our hydroponics application.    But first we’re going to have to re-visit the two T’s of the Euler-Eytelwein Formula.   We’ll do that next time.

 

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Another Specialized Application of the Euler-Eyelewein Formula

Tuesday, June 13th, 2017

    Last week we saw how friction coefficients as used in the Euler-Eyelewein Formula, can be highly specific to a specialized application, U.S. Navy ship capstans.   In fact, many diverse industries benefit from aspects of the Euler-Eytelwein Formula.   Today we’ll introduce another engineering application of the Formula, exploring its use within the irrigation system of a hydroponics plant.

Another Specialized Application of the Euler-Eyelewein Formula

Another Specialized Application of the Euler-Eyelewein Formula

   

    Pumps conveying water are an indispensable part of a hydroponics plant.   In the schematic shown here they are portrayed by the symbol ⊗.

    In our simplified scenario to be presented next week, these pumps are powered by a mechanical power transmission system, each consisting of two pulleys and a belt.   One pulley is connected to a water pump, the other pulley to a gasoline engine.   A belts runs between the pulleys to deliver mechanical power from the engine to the pump.

    The width of the belts is a key component in an efficiently running hydroponics plant.   We’ll see how and why that’s so next time.

 

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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The Two T’s of the Euler-Eytelwein Formula

Sunday, May 14th, 2017

    Last time we introduced some of the variables in the Euler-Eytelwein Formula, an equation used to examine the amount of friction present in pulley-belt assemblies.   Today we’ll explore its two tension-denoting variables, T1 and T2.

   Here again is the Euler-Eytelwin Formula,where, T1 and  T2 are belt tensions on either side of a pulley,

T1 = T2 × eθ)

    T1 is known as the tight side tension of the assembly because, as its name implies, the side of the belt containing this tension is tight, and that is so due to its role in transmitting mechanical power between the driving and driven pulleys.   T2 is the slack side tension because on its side of the pulley no mechanical power is transmitted, therefore it’s slack–it’s just going along for the ride between the driving and driven pulleys.

   Due to these different roles, the tension in T1 is greater than it is in T2.

The Two T’s of the Euler-Eytelwein Formula

The Two T’s of the Euler-Eytelwein Formula

    In the illustration above, tension forces T1 and T2 are shown moving in the same direction, because the force that keeps the belt taught around the pulley moves outward and away from the center of the pulley.

    According to the Euler-Eytelwein Formula, T1 is equal to a combination of factors:  tension T2 ; the friction that exists between the belt and pulley, denoted as μ; and how much of the belt is in contact with the pulley, namely θ.

    We’ll get into those remaining variables next time.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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A First Look at the Euler-Eytelwein Formula

Friday, May 5th, 2017

    Last time we introduced the Pulley Speed Ratio Formula, a Formula which assumes a certain amount of friction in a pulley-belt assembly in order to work.   Today we’ll introduce another Formula, one which oversees how friction comes into play between belts and pulleys, the Euler-Eytelwein Formula.   It’s a Formula developed by two pioneers of engineering introduced in an earlier blog, Leonhard Euler and Johann Albert Eytelwein.

    Here again is the Pulley Speed Ratio Formula,

D1 × N1 = D2 × N2

where, D1 is the diameter of the driving pulley and D2 the diameter of the driven pulley. The pulleys’ rotational speeds are represented by N1 and N2.

   This equation works when it operates under the assumption that friction between the belt and pulleys is, like Goldilock’s preferred bed, “just so.”   Meaning, friction present is high enough so the belt doesn’t slip, yet loose enough so as not to bring the performance of a rotating piece of machinery to a grinding halt.

    Ideally, you want no slippage between belt and pulleys, but the only way for that to happen is if you have perfect friction between their surfaces—something that will never happen because there’s always some degree of slippage.   So how do we design a pulley-belt system to maximize friction and minimize slip?

    Before we get into that, we must first gain an understanding of how friction comes into play between belts and pulleys.   To do so we’ll use the famous Euler-Eytelwein Formula, shown here,

A First Look at the Euler-Eytelwein Formula

A First Look at the Euler-Eytelwein Formula

   

where, T1 and  T2 are belt tensions on either side of a pulley.

    We’ll continue our exploration of the Euler-Eytelwein Formula next time when we discuss the significance of its two sources of tension.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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A Pulley Speed Ratio Formula Application

Friday, April 21st, 2017

    Last time we saw how pulley diameter governs speed in engineering scenarios which make use of a belt and pulley system.   Today we’ll see how this phenomenon is defined mathematically through application of the Pulley Speed Ratio Formula, which enables precise pulley diameters to be calculated to achieve specific rotational speeds.   Today we’ll apply this Formula to a scenario involving a building’s ventilating system.

    The Pulley Speed Ratio Formula is,

                                                D1 × N1 =  D2 × N2                             (1)

where, D1 is the diameter of the driving pulley and D2 the diameter of the driven pulley.

A Pulley Speed Ratio Formula Application

A Pulley Speed Ratio Formula Application

   

The pulleys’ rotational speeds are represented by N1 and N2,  and are measured in revolutions per minute (RPM).

    Now, let’s apply Equation (1) to an example in which a blower must deliver a specific air flow to a building’s ventilating system.   This is accomplished by manipulating the ratios between the driven pulley’s diameter, D2, with respect to the driving pulley’s diameter, D1.   If you’ll recall from our discussion last time, when both the driving and driven  pulleys have the same diameter, the entire assembly moves at the same speed, and this would be bad for our scenario.

    An electric motor and blower impeller moving at the same speed is problematic because electric motors are designed to spin at much faster speeds than typical blower impellers in order to produce desired air flow.   If their pulleys’ diameters were the same size, it would result in an improperly working ventilating system in which air passes through the furnace heat exchanger and air conditioner cooling coils far too quickly to do an efficient job of heating or cooling.

    To bear this out, let’s suppose we have an electric motor turning at a fixed speed of 3600 RPM and a belt-driven blower with an impeller that must turn at 1500 RPM to deliver the required air flow according to the blower manufacturer’s data sheet.   The motor shaft is fitted with a pulley 3 inches in diameter.   What pulley diameter do we need for the blower to turn at the manufacturer’s required 1500 RPM?

    In this example known variables are D1 = 3 inches, N1 = 3600 RPM, and N2 = 1500 RPM.   The diameter D2 is unknown.   Inserting the known values into equation (1), we can solve for D2,

                               (3 inches) × (3600 RPM) = D2 × (1500 RPM)         (2)

Simplified, this becomes,

                                            D2 = 7.2 inches                                      (3)

    Next time we’ll see how friction affects our scenario.

   

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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The Difference Between Driven and Driving Pulleys

Friday, March 31st, 2017

    Last time we introduced two historical legends in the field of engineering who pioneered the science of mechanical power transmission using belts and pulleysLeonhard Euler and Johann Albert Eytelwein.   Today we’ll build a foundation for understanding their famous Euler-Eytelwein Formula through our example of a simple mechanical power transmission system consisting of two pulleys and a belt, and in so doing demonstrate the difference between driven and driving pulleys.

    Our example of a basic mechanical power transmission system consists of two pulleys connected by a drive belt.   The driving pulley is attached to a source of mechanical power, for example, the shaft of an electric motor.   The driven pulley, which is attached to the shaft of a piece of rotating machinery, receives the mechanical power from the electric motor so the machinery can perform its function.

The Difference Between Driven and Driving Pulleys

The Difference Between Driven and Driving Pulleys 

   

    Next time we’ll see how driven pulleys can be made to spin at different speeds from the driving pulley, enabling different modes of operation in mechanical devices.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Johann Albert Eytelwein, Engineering Trailblazer

Monday, March 20th, 2017

    They say necessity is the mother of invention, and today’s look at an influential historical figure in engineering bears that out.   Last week we introduced Leonhard Euler and touched on his influence to the science of pulleys.   Today we’ll introduce his contemporary and partner in science, Johann Albert Eytelwein, a German mathematician and visionary, a true engineering trailblazer whose contributions to the blossoming discipline of engineering led to later studies with pulleys.

 Johann Albert Eytelwein, Engineering Trailblazer

Johann Albert Eytelwein, Engineering Trailblazer

   

    Johann Albert Eytelwein’s experience as a civil engineer in charge of the dikes of former Prussia led him to develop a series of practical mathematical problems that would enable his subordinates to operate more effectively within their government positions.   He was a trailblazer in the field of applied mechanics and their application to physical structures, such as the dikes he oversaw, and later to machinery.   He was instrumental in the founding of Germany’s first university level engineering school in 1799, the Berlin Bauakademie, and served as director there while lecturing on many developing engineering disciplines of the time, including machine design and hydraulics.   He went on to publish in 1801 one of the most influential engineering books of his time, entitled Handbuch der Mechanik (Handbook of the Mechanic), a seminal work which combined what had previously been mere engineering theory into a means of practical application.

    Later, in 1808, Eytelwein expanded upon this work with his Handbuch der Statik fester Koerper (Handbook of Statics of Fixed Bodies), which expanded upon the work of Euler.   In it he discusses friction and the use of pulleys in mechanical design.  It’s within this book that the famous Euler-Eytelwein Formula first appears, a formula Eytelwein derived in conjunction with Euler.   The formula delves into the usage of belts with pulleys and examines the tension interplay between them.

    More on this fundamental foundation to the discipline of engineering next time, with a specific focus on pulleys.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Leonhard Euler, a Historical Figure in Pulleys

Thursday, March 9th, 2017

    Last time we ended our blog series on pulleys and their application within engineering as aids to lifting.   Today we’ll embark on a new focus series, pulleys used in mechanical devices.   We begin with some history, a peek at Swiss scientist and mathematician Leonhard Euler, a historical figure credited to be perhaps the greatest mathematician of the 18th Century.

   

Leonhard Euler, a Historical Figure in Pulleys

Leonhard Euler, a Historical Figure in Pulleys

   

    Euler is so important to math, he actually has two numbers named after him.   One is known simply as Euler’s Number, 2.7182, most often notated as e, the other Euler’s Constant, 0.57721, notated γ, which is a Greek symbol called gamma.   In fact, he developed most math notations still in use today, including the infamous function notation, f(x), which no student of elementary algebra can escape becoming intimately familiar with.

    Euler authored his first theoretical essays on the science and mathematics of pulleys after experimenting with combining them with belts in order to transmit mechanical power.   His theoretical work became the foundation of the formal science of designing pulley and belt drive systems.   And together with German engineer Johann Albert Eytelwein, Euler is credited with a key formula regarding pulley-belt drives, the Euler-Eytelwein Formula, still in use today, and which we’ll be talking about in depth later in this blog series.

    We’ll talk more about Eytelwein, another important historical figure who worked with pulleys, next time.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Mechanical Power Transmission – Putting the Centrifugal Clutch Together

Sunday, April 29th, 2012

     I’ve never been one to enjoy table top puzzles, yet I love to examine the way mechanical things fit together.  Manipulating parts to see how they interrelate to form an operational system is a pastime I very much enjoy.  In fact, I spend many evenings at my work bench doing just this.  I often become so engrossed in the activity I forget what time it is.  The result is yet another night without TV.  So sad…

     Last week we looked at how a centrifugal clutch mechanism operates when it’s coupled to a gasoline engine shaft spinning at idle speed, and then we depressed the engine throttle trigger to speed things up.  Let’s now introduce a new component called the clutch housing to see how it interfaces with the clutch mechanism to drive the cutter head in a grass trimmer.

centrifugal clutch housing

Figure 1

 

     The clutch housing shown in Figure 1 resembles a rather short cup.  One end is open, the other closed.

     Figure 2 shows the closed end of the clutch housing connected to the cutter shaft’s coupling.  On the cutter shaft coupling resides a ball bearing which enables the clutch housing to both spin freely and act as a support for the clutch housing.  The open end of the clutch housing allows the clutch mechanism to fit neatly inside.

centrifugal clutch assembly

Figure 2

 

     Next time we’ll put the assembly shown in Figure 2 into operation.  First we’ll examine how the centrifugal clutch mechanism and clutch housing operate with the engine at idle speed, then compare that to the engine operating at actual cutting speed.

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